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2. Measurement Process Characterization
2.3. Calibration
2.3.4. Catalog of calibration designs

Mass weights

Tie to kilogram reference standards Near-accurate mass measurements require a sequence of designs that relate the masses of individual weights to a reference kilogram(s) standard ( Jaeger & Davis). Weights generally come in sets, and an entire set may require several series to calibrate all the weights in the set.
Example of weight set A 5,3,2,1 weight set would have the following weights:

1000 g

500g, 300g, 200g, 100g

50g, 30g 20g, 10g

5g, 3g, 2g, 1g

0.5g, 0.3g, 0.2g, 0.1g

Depiction of a design with three series for calibrating a 5,3,2,1 weight set with weights between 1 kg and 10 g
depiction of a design for calibrating weights
First series using 1,1,1,1 design The calibrations start with a comparison of the one kilogram test weight with the reference kilograms (see the graphic above). The 1,1,1,1 design requires two kilogram reference standards with known values, \( R_1^* \) and \( R_2^* \). The fourth kilogram in this design is actually a summation of the 500, 300, 200 g weights which becomes the restraint in the next series.

The restraint for the first series is the known average mass of the reference kilograms, $$ R^* = \frac{R_1^* + R_2^*}{2} \, .$$ The design assigns values to all weights including the individual reference standards. For this design, the check standard is not an artifact standard but is defined as the difference between the values assigned to the reference kilograms by the design; namely, $$ C = \left( \widehat{R_1^*} - \widehat{R_2^*} \right) \, . $$

Second series using 5,3,2,1,1,1 design The second series is a 5,3,2,1,1,1 design where the restraint over the 500g, 300g and 200g weights comes from the value assigned to the summation in the first series; i.e., $$ R^* = \sum 1 = \widehat{X}_{500} + \widehat{X}_{300} + \widehat{X}_{200} \, .$$ The weights assigned values by this series are:
  • 500g, 300g, 200 g and 100g test weights
  • 100 g check standard (2nd 100g weight in the design)
  • Summation of the 50g, 30g, 20g weights.
Other starting points The calibration sequence can also start with a 1,1,1 design. This design has the disadvantage that it does not have provision for a check standard.
Better choice of design A better choice is a 1,1,1,1,1 design which allows for two reference kilograms and a kilogram check standard which occupies the 4th position among the weights. This is preferable to the 1,1,1,1 design but has the disadvantage of requiring the laboratory to maintain three kilogram standards.
Important detail The solutions are only applicable for the restraints as shown.
Designs for decreasing weight sets
  1. 1,1,1 design
  2. 1,1,1,1 design
  3. 1,1,1,1,1 design
  4. 1,1,1,1,1,1 design
  5. 2,1,1,1 design
  6. 2,2,1,1,1 design
  7. 2,2,2,1,1 design
  8. 5,2,2,1,1,1 design
  9. 5,2,2,1,1,1,1 design
  10. 5,3,2,1,1,1 design
  11. 5,3,2,1,1,1,1 design
  12. 5,3,2,2,1,1,1 design
  13. 5,4,4,3,2,2,1,1 design
  14. 5,5,2,2,1,1,1,1 design
  15. 5,5,3,2,1,1,1 design
  16. 1,1,1,1,1,1,1,1 design
  17. 3,2,1,1,1 design
Design for pound weights
  1. 1,2,2,1,1 design
Designs for increasing weight sets
  1. 1,1,1 design
  2. 1,1,1,1 design
  3. 5,3,2,1,1 design
  4. 5,3,2,1,1,1 design
  5. 5,2,2,1,1,1 design
  6. 3,2,1,1,1 design
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